To illustrate this method, we will compute the square root of two. Now,
we cannot use the equation x2-2=0 because it has two roots,
+2 and -2, which are equal in absolute value. So what
we shall do instead is to use the equation (x-1)2=2, or x2-2⁢x-3=0, whose roots are 1-2 and 1+2. Since
|1+2|>|1-2|, we can use our method to approximate
the larger of these roots, namely 1+2; to approximate 2,
we subtract 1 from our answer. The recursion we should use is

By making suitable transformations, one can compute all the roots
of a polynomial using this technique. A way to do this is to start
with a rational number h which is closer to the desired root than
to the other roots, then make a change of variable x=1/(y+h).

As an example, we shall examine the roots of x3+9⁢x+1.
Approximating the polynomial by leaving off the constant term, we
guess that the roots are close to 0, +3⁢i, and -3⁢i. Since the
two complex roots are conjugates, it suffices to find one of them.

To look for the root near 0, we make the change of variable x=1/y to obtain y3+9⁢y2+1=0, which gives the recursion
ak+3+9⁢ak+2+ak=0. Picking some initial values, this
recursion gives us the sequence

0,0,1,-9,81,-738,6723,-61236,557766,-5090401,…,

whence we obtain the approximations

-19,-19,-81738,-7386723,-672361236,-61236557766,-5577665090401,….

To look for the root near 3⁢i, we make the change of variable
x=1/(y+3⁢i) to obtain y3+(9+9⁢i)⁢y2+(-27+54⁢i)⁢y+82-27⁢i=0, which gives the recursion